Transformations of e-Biased Distributions

Abhijit Sahay

Most approaches to reducing (or eliminating) the use of randomness in probabilistic algorithms can be viewed as attempts to replace the probability space from which the algorithm samples by a much smaller one. The use of limited independence among the random choices of the algorithm has long been used as a technique for constructing small spaces that "approximate" the exponential-sized spaces resulting from complete independence. Naor & Naor suggested the use of e-biased spaces as an alternative approximation scheme.

We present results that exhibit severe limitations in the extent to which e-biased spaces can approximate unbiased ones. Specifically, we show that the small bias of an e-biased space can be destroyed by extremely simple transformations, thereby rendering such spaces unsuitable for any randomized algorithm that employs such transformations. Our results partially explain the paucity of applications for e-biased spaces as well as of tools (such as probability tail bounds) for analyzing these distributions. They also suggest that this state of affairs is likely to persist because of the inherent fragility of the notion of bias of a sample space.

BibTeX citation:

@techreport{Sahay:CSD-92-710,
Author = {Sahay, Abhijit},
Title = {Transformations of e-Biased Distributions},
Institution = {EECS Department, University of California, Berkeley},
Year = {1992},
Month = {Nov},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1992/6264.html},
Number = {UCB/CSD-92-710},
Abstract = {Most approaches to reducing (or eliminating) the use of randomness in probabilistic algorithms can be viewed as attempts to replace the probability space from which the algorithm samples by a much smaller one. The use of limited independence among the random choices of the algorithm has long been used as a technique for constructing small spaces that "approximate" the exponential-sized spaces resulting from complete independence. Naor & Naor suggested the use of e-biased spaces as an alternative approximation scheme. <p>We present results that exhibit severe limitations in the extent to which e-biased spaces can approximate unbiased ones. Specifically, we show that the small bias of an e-biased space can be destroyed by extremely simple transformations, thereby rendering such spaces unsuitable for any randomized algorithm that employs such transformations. Our results partially explain the paucity of applications for e-biased spaces as well as of tools (such as probability tail bounds) for analyzing these distributions. They also suggest that this state of affairs is likely to persist because of the inherent fragility of the notion of bias of a sample space.}
}